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\title{Math 420-620\\Fall 2012\\Homework 7}
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\centerline{\it{Due Wednesday, October 17, 2012}}

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\noindent 1. (5 pt) Let $H$ be a subgroup of $G$. Show that $\bigcap_{x\in G} xHx^{-1}$ is always normal in $G$.

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\noindent 2. Let $G$ be a group and $H$ a subgroup. We say that $H$ is characteristic in $G$ if $\phi(H)\subseteq H$ for all $\phi\in\text{Aut}(G)$.
\begin{itemize}
\item[a)] (5 pt) Show that if $H$ is characteristic in $G$, then $H$ is normal in $G$.
\item[b)] (5 pt) Show that an arbitrary intersection of characteristic subgroups of $G$ is characteristic.
\item[c)] (5 pt) Give an example of a group $G$ with a normal subgroup that is not characteristic.
\item[d)] (5 pt) Show that $Z(G)$ is a characteristic subgroup of $G$.
\end{itemize}

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\noindent 3. (5 pt) Let $G$ be a group and let $G$ act on itself by conjugation, that is, $g\cdot x=gxg^{-1}$.
\begin{itemize}
\item[a)] (5 pt) Show that the above is, in fact, a group action.
\item[b)] (5 pt) What is the kernal of this action?
\item[c)] (5 pt) Show that if $a\in G$ then $G_a=C_G(a)$.
\end{itemize}

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\noindent 4. (5 pt) Let $G$ be a group. Show that $\text{Inn}(G)\cong G/Z(G)$.




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